The Sixth Painlevé Equation as Similarity Reduction of $${\widehat{\mathfrak{gl}}_3}$$ Generalized Drinfel’d–Sokolov Hierarchy

Kakei, Saburo; Kikuchi, Tetsuya

doi:10.1007/s11005-007-0144-4

Cited by 14 publications

(15 citation statements)

References 23 publications

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“…The one-dimensional restriction of the similarity reduction (3.1) was used in the recent works [8] and [21] to obtain a 3 × 3 Fuchs-Garnier pair for P 6 from the (1+1)-dimensional scattering Lax pair. In the remainder of this section we will rederive this result from the (2+1)-dimensional perspective.…”

Section: Similarity Reduction To the Sixth Painlevé Equationmentioning

confidence: 99%

“…Recently, there appeared two independent works by R. Conte, A. M. Grundland, and M. Musette [8] and S. Kakei, T. Kikuchi [21] were the authors obtain Harnad's FuchsGarnier pair for P 6 by using an extension of the similarity reduction [25] for the three-wave resonant interaction (3WRI) system in (1 + 1) (one spatial and one time) dimensions, to the corresponding Lax pair. The Lax pair for this system was given in terms of two commuting first order differential operators in 3 × 3 matrices by V. E. Zakharov and S. V. Manakov [42].…”

Section: Introductionmentioning

confidence: 99%

“…The Lax pair for this system was given in terms of two commuting first order differential operators in 3 × 3 matrices by V. E. Zakharov and S. V. Manakov [42]. The authors of [8] and [21] were able to get the Fuchs-Garnier pair already studied by Harnad and Mazzocco and used their parametrization to get explicit formulae for the solutions of 3WRI system in terms of P 6 with the complete set of the coefficients.…”

Section: Introductionmentioning

confidence: 99%

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On the linearization of the Painlevé III–VI equations and reductions of the three-wave resonant system

Joshi

Kitaev

Treharne

2007

Journal of Mathematical Physics

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We extend similarity reductions of the coupled (2+1)-dimensional three-wave resonant interaction system to its Lax pair. Thus we obtain new 3 × 3 matrix Fuchs-Garnier pairs for the third, fourth, and fifth Painlevé equations, together with the previously known Fuchs-Garnier pair for the sixth Painlevé equation. These Fuchs-Garnier pairs have an important feature: they are linear with respect to the spectral parameter. Therefore we can apply the Laplace transform to study these pairs. In this way we found reductions of all pairs to the standard 2 × 2 matrix Fuchs-Garnier pairs obtained by M. Jimbo and T. Miwa. As an application of the 3 × 3 matrix pairs, we found an integral auto-transformation for the standard Fuchs-Garnier pair for the fifth Painlevé equation. It generates an Okamoto-like Bäcklund transformation for the fifth Painlevé equation. Another application is an integral transformation relating two different 2 × 2 matrix Fuchs-Garnier pairs for the third Painlevé equation.

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Section: Similarity Reduction To the Sixth Painlevé Equationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On the linearization of the Painlevé III–VI equations and reductions of the three-wave resonant system

Joshi

Kitaev

Treharne

2007

Journal of Mathematical Physics

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“…The Drinfeld-Sokolov hierarchies are extensions of the KdV (or mKdV) hierarchy for the a‰ne Lie algebras [DS]. For type A ð1Þ n , they imply several Painlevé systems by similarity reductions [AS,KIK,KK1,KK2,NY1]; see Table 1. Such fact clarifies the origins of several properties of the Painlevé systems, Lax pairs, a‰ne Weyl group symmetries and particular solutions in terms of the Schur polynomials.…”

Section: Introductionmentioning

confidence: 99%

“…. ; 1Þ of n þ 2, we have the Garnier system in n-variables [KK2]. Also for each partition ð5; 1Þ and ð2; 2; 2Þ, a system of sixth order is derived; we do not give the explicit formula here.…”

Section: Introductionmentioning

confidence: 99%

Drinfeld-Sokolov Hierarchies of Type A and Fourth Order Painleve Systems

Fuji

Suzuki

2010

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From the Darboux–Egorov system to bi-flat F-manifolds

Arsie

Lorenzoni

2013

Journal of Geometry and Physics

137

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Motivated by the theory of integrable PDEs of hydrodynamic type and by the generalization of Dubrovin's duality in the framework of F -manifolds due to Manin [22], we consider a special class of F -manifolds, called bi-flat F -manifolds.A bi-flat F -manifold is given by the following data (M, ∇ 1 , ∇ 2 , •, * , e, E), where (M, •) is an F -manifold, e is the identity of the product •, ∇ 1 is a flat connection compatible with • and satisfying ∇ 1 e = 0, while E is an eventual identity giving rise to the dual product * , and ∇ 2 is a flat connection compatible with * and satisfying ∇ 2 E = 0. Moreover, the two connections ∇ 1 and ∇ 2 are required to be hydrodynamically almost equivalent in the sense specified in [2].First we show that, similarly to the way in which Frobenius manifolds are constructed starting from Darboux-Egorov systems, also bi-flat F -manifolds can be built from solutions of suitably augmented Darboux-Egorov systems, essentially dropping the requirement that the rotation coefficients are symmetric.Although any Frobenius manifold possesses automatically the structure of a bi-flat F -manifold, we show that the latter is a strictly larger class.In particular we study in some detail bi-flat F -manifolds in dimensions n = 2, 3. For instance, we show that in dimension three bi-flat F -manifolds are parametrized by solutions of a two parameters Painlevé VI equation, admitting among its solutions hypergeometric functions. Finally we comment on some open problems of wide scope related to bi-flat F -manifolds.

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The Sixth Painlevé Equation as Similarity Reduction of $${\widehat{\mathfrak{gl}}_3}$$ Generalized Drinfel’d–Sokolov Hierarchy

Cited by 14 publications

References 23 publications

On the linearization of the Painlevé III–VI equations and reductions of the three-wave resonant system

On the linearization of the Painlevé III–VI equations and reductions of the three-wave resonant system

Drinfeld-Sokolov Hierarchies of Type A and Fourth Order Painleve Systems

From the Darboux–Egorov system to bi-flat F-manifolds

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